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We consider the Dirichlet problem for solutions to general second-orderhomogeneous elliptic equations with constant complex coefficients. We provethat any Jordan domain with $C^{1,\alpha}$-smooth boundary, $0<\alpha<1$, isnot regular with respect to the Dirichlet problem for any not strongly ellipticequation $\mathcal Lf=0$ of this kind, which means that for any such domain $G$it always exists a continuous function on the boundary of $G$ that can not becontinuously extended to the domain under consideration to a functionsatisfying the equation $\mathcal Lf=0$ therein. Since there exists a Jordandomain with Lipschitz boundary that is regular with respect to the Dirichletproblem for bianalytic functions, this result is near to be sharp. We alsoconsider several connections between Dirichlet problem for elliptic equationsunder consideration and problems on uniform approximation by polynomialsolutions of such equations.

On Dirichlet problem for second-order elliptic equations in the plane and uniform approximation problems for solutions of such equations